Yes, I mean the explicit Theta-space model. Thanks for the reply below! I could now keep asking questions along the lines: "How much category theory is known for Theta-spaces"? Is there an explicit definition of limits and colimits, for instance? Of adjoint functors?
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Urs SchreiberNov 18 '09 at 7:41

1 Answer
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The totality of $r$-$\Theta$-spaces forms a (large) category enriched over $r$-$\Theta$-spaces, which I'll call $C$. Given this, you can form a presheaf of spaces $X$ on the category $\Theta_{r+1}$, where

Here "$\[m\](\theta_1,\dots,\theta_m)$" represents a typical object in $\Theta_{r+1}$ (so each $\theta_i\in \Theta_r$). The coproduct is over tuples of objects of $C$.
The structure maps in the presheaf use the fact that $C$ is a category object. (It's like the way you get a Segal category from a category enriched over spaces.)

The gadget $X$ is almost an $(r+1)$-$\Theta$-space. It satisfies all the "Segal" conditions, and also all the completeness conditions except for the one in bottom dimension. You get an honest $(r+1)$-$\Theta$-space $X'$ from $X$ by applying a suitable localization.

The gadget $X'$ should be the thing you want. (None of the proofs involved here have been written up, or at least not by me, though we're working on it.)